Self-adjoint extensions for a $p^{4}$-corrected Hamiltonian of a particle on a finite interval

TL;DR

This paper characterizes all self-adjoint extensions of a $p^4$-corrected Hamiltonian on a finite interval, facilitating quantum models at minimal length scales relevant to quantum gravity.

Contribution

It provides an explicit $U(4)$ parametrization of self-adjoint extensions for the $p^4$-corrected Hamiltonian on a finite interval, enabling modeling of various boundary conditions.

Findings

Explicit $U(4)$ parametrization of self-adjoint extensions.

Application to different boundary conditions like infinite square-well and periodic.

Framework for modeling quantum systems with minimal-length scale corrections.

Abstract

In the present paper we deal with the issue of finding the self-adjoint extensions of a $p^4$-corrected Hamiltonian. The importance of this subject lies on the application of the concepts of quantum mechanics to the minimal-length scale scenario which describes an effective theory of quantum gravity. We work in a finite one dimensional interval and we give the explicit $U(4)$ parametrization that leads to the self-adjoint extensions. Once the parametrization is known, we can choose appropriate $U(4)$ matrices to model physical problems. As examples, we discuss the infinite square-well, periodic conditions, anti-periodic conditions and periodic conditions up to a prescribed phase. We hope that the parametrization we found will contribute to model other interesting physical situations in further works.